Question: Simplify; express your answer in exponential form. Assume $z\neq 0, y\neq 0$. $\dfrac{{(z^{5}y^{3})^{3}}}{{(z^{-4}y^{-1})^{-1}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{5}y^{3})^{3} = (z^{5})^{3}(y^{3})^{3}}$ On the left, we have ${z^{5}}$ to the exponent ${3}$ . Now ${5 \times 3 = 15}$ , so ${(z^{5})^{3} = z^{15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{5}y^{3})^{3}}}{{(z^{-4}y^{-1})^{-1}}} = \dfrac{{z^{15}y^{9}}}{{z^{4}y}}$ Break up the equation by variable and simplify. $\dfrac{{z^{15}y^{9}}}{{z^{4}y}} = \dfrac{{z^{15}}}{{z^{4}}} \cdot \dfrac{{y^{9}}}{{y}} = z^{{15} - {4}} \cdot y^{{9} - {1}} = z^{11}y^{8}$